** Title:** The Hodge theory of algebraic maps

** Abstract:** I will report on my joint work with Luca Migliorini at Bologna
on the homology of algebraic maps. We give a geometric proof of the
so-called Decomposition Theorem due to Beilinson-Bernstein-Deligne and
Gabber concerning the relation between the homology theories on
the domain and target of an algebraic map. Our approach identifies
the non degeneration of certain intersection forms on the homology
of the fibers of the map as the reason for the decomposition
of the homology of the domain into certain pieces defined on
the target. This fact has no counterpart in other geometries, e.g.
complex geometry, real algebraic geometry. I will
illustrate the discussion with some key elementary examples.